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Let $G$ be a finite group of order $n>1$. Show that $|Aut($G$)|<n^{\log_2(n)}$.

The 'only' thing I know is the group of inner automorphisms is isomorphic to $G/Z(G)$ and by definition, $Z(G)=G$ if and only if $G$ is abelian.

If $|G|>2$ this thread show that Aut($G$) contains at least two elements.

But how can I have an upper bound ?

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Here are some hints - more details on request.

1.Show first that any finite group of order $n$ has a generating set $X$ with $|X| \le \log_2(n)$.

2.Show that any homomorphism $\phi$ with domain $G$ is completely determined by the images of $\phi$ on a generating set $X$.

3.Deduce the result, taking care to prove the strict inequality.

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  • $\begingroup$ I have seen the question/hint (1) before on this site ... $\endgroup$ – Nicky Hekster Jun 25 '14 at 10:17
  • $\begingroup$ Thanks for hints and not a full solution, now I can work by myself! :) $\endgroup$ – user146010 Jun 25 '14 at 10:47

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